Answer
$ a.\quad$ critical points at $x=0$ and $x=3$
$ b.\quad$ f is increasing on $(3, \infty)$, decreasing on $(0,3)$.
$ c.\quad$ local minimum at $x=3$
Work Step by Step
$(a)$
$f'$ is not defined at $ x=0\qquad$ ... critical point
(x=0 has not been excluded, so we take that f is defined at x=0)
$f'(x)=0$ for $ x=3\qquad$ ... critical point
Critical points at $x=0$ and $x=3$
$(b)$
$\left[\begin{array}{cccc}
& (0,3) & (3,\infty) & \\
\text{test point, }t & 1 & 4 & \\
\text{evaluate }f'(t) & -2 & 1/2 &
\end{array}\right]$
$f':\qquad \stackrel{0}{} \stackrel{\searrow}{- - -} \stackrel{3}{|} \stackrel{\nearrow}{+++}$
f is increasing on $(3, \infty)$, decreasing on $(0,3)$.
$(c)$
From the table, we see that f has a local minimum at $x=3$
